Catalysts comprising palladium and an inorganic support are known catalysts for the hydrogenation of polyenes and/or alkynes. Even though these catalysts are effective hydrogenation catalysts, they ten to produce green oil by oligomerizing alkenes, polyenes, and/or alkynes. Green oil, as used herein, refers to molecules having 6 or more carbon atoms per molecule and is undesirable in the production of an alkene because it fouls the hydrogenation catalyst which in turn deactivates the catalyst. The deactivation process can lower the activity and selectivity of the catalyst. Therefore, there is an ever present need for further improvements in the selective hydrogenation process for converting a highly unsaturated hydrocarbon to a less unsaturated hydrocarbon and to achieve enhanced selectivity to the less unsaturated hydrocarbon, or increased catalyst life, or both. Accordingly, the development of a modified supported palladium catalyst composition and its use in processes for the selective hydrogenation of highly unsaturated hydrocarbons such as diolefins (alkadienes) or alkynes to less unsaturated hydrocarbons such as monoolefins (alkenes) would be a significant contribution to the art.